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A KOSZUL DUALITY for PROPS Introduction the Koszul Duality Is a Theory Developed for the First Time in 1970 by S. Priddy For TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 10, October 2007, Pages 4865–4943 S 0002-9947(07)04182-7 Article electronically published on May 16, 2007 AKOSZULDUALITYFORPROPS BRUNO VALLETTE Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props. Introduction The Koszul duality is a theory developed for the first time in 1970 by S. Priddy for associative algebras [Pr]. To every quadratic algebra A, it associates a dual coalgebra A¡ and a chain complex called a Koszul complex. When this complex is acyclic, we say that A is a Koszul algebra. In this case, the algebra A and its representations have many properties (cf. A. Beilinson, V. Ginzburg and W. Soergel [BGS]). In 1994, this theory was generalized to algebraic operads by V. Ginzburg and M.M. Kapranov (cf. [GK]). An operad is an algebraic object that models the operations with n inputs (and one output) A⊗n → A acting on a type of algebras. For instance, there exist operads As, Com and Lie coding associative, commutative and Lie algebras. The Koszul duality theory for operads has many applications: construction of a “small” chain complex to compute the homology groups of an algebra, minimal model of an operad, and notion of algebra up to homotopy. The discovery of quantum groups (cf. V. Drinfeld [Dr1, Dr2]) has popularized algebraic structures with products and coproducts, that’s to say, operations with multiple inputs and multiple outputs A⊗n → A⊗m . It is the case of bialgebras, Hopf algebras and Lie bialgebras, for instance. The framework of operads is too narrow to treat such structures. To model the operations with multiple inputs and outputs, one has to use a more general algebraic object: the props. Following J.-P. Serre in [S], we call an “algebra” over a prop P,aP-algebra. It is natural to try to generalize Koszul duality theory to props. A few works have been done in that direction by M. Markl and A.A. Voronov in [MV] and W.L. Gan in [G]. Actually, M. Markl and A.A. Voronov proved Koszul duality theory 1 for what M. Kontsevich calls 2 -PROPs, and W.L. Gan proved it for dioperads. One has to bear in mind that an associative algebra is an operad, an operad is a 1 1 2 -PROP, a 2 -PROP is a dioperad and a dioperad induces a prop: 1 Associative algebras → Operads → − PROPs → Dioperads ; Props. 2 Received by the editors June 27, 2005. 2000 Mathematics Subject Classification. Primary 18D50; Secondary 16W30, 17B26, 55P48. Key words and phrases. Prop, Koszul duality, operad, Lie bialgebra, Frobenius algebra. c 2007 American Mathematical Society Reverts to public domain 28 years from publication 4865 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4866 BRUNO VALLETTE In this article, we introduce the notion of properad, which is the connected part of a prop and contains all the information to code algebraic structure like bialgebras, Lie bialgebras, and infinitesimal Hopf algebras (cf. M. Aguiar [Ag1], [Ag2], [Ag3]). We prove the Koszul duality theory for properads which gives Koszul duality for quadratic props defined by connected relations (it is the case of the two last examples). The first difficulty is to generalize the bar and cobar constructions to proper- ads and props. The differentials of the bar and cobar constructions for associative 1 algebras, operads, 2 -PROPs and dioperads are given by the notions of ‘edge con- traction’ and ‘vertex expansion’ introduced by M. Kontsevich in the context of graph homology (cf. [Ko]). We define the differentials of the bar and cobar con- structions of properads using conceptual properties. This generalizes the previous constructions. The preceding proofs of Koszul duality theories are based on combinatorial prop- erties of trees or graphs of genus 0. Such proofs cannot be used in the context of props since we have to work with any kind of graph. To solve this problem, we intro- duce an extra grading induced by analytic functors and generalize the comparison lemmas of B. Fresse [Fr] to properads. To any quadratic properad P, we associated a Koszul dual coproperad P¡ and a Koszul complex. The main theorem of this paper is a criterion which claims that the Koszul complex is acyclic if and only if the cobar construction of the Koszul dual P¡ is a resolution of P. In this case, we say that the properad is Koszul and this resolution is the minimal model of P. As a corollary, we get the deformation theories for gebras over Koszul properads. For instance, we prove that the properad of Lie bialgebras and the properad of infinitesimal Hopf algebras are Koszul, which gives the notions of Lie bialgebra up to homotopy and infinitesimal Hopf algebra up to homotopy. Structures of gebras up to homotopy appear in the level of on some chain complexes. For instance, the Hochschild cohomology of an associative algebra with coefficients into itself has a structure of Gerstenhaber algebra up to homotopy (cf. J.E. McClure and J.H. Smith [McCS], see also C. Berger and B. Fresse [BF]). This conjecture is known as Deligne’s conjecture. The same kind of conjectures exist for gebras with coproducts up to homotopy. One of them was formulated by M. Chas and D. Sullivan in the context of String Topology (cf. [CS]). This conjecture says that the structure of involutive Lie bialgebra (a particular type of Lie bialgebras) on some homology groups can be lifted to a structure of involutive Lie bialgebras up to homotopy. Working with complexes of graphs of genus greater than 0 is not an easy task. This Koszul duality for props provides new methods for studying the homology of these chain complexes. One can interpret the bar and cobar constructions of Koszul properads in terms of graph complexes. Therefore, it is possible to compute their (co)homology in Kontsevich’s sense. For instance, the case of the properad of Lie bialgebras leads to the computation of the cohomology of classical graphs, and the case of the properad of infinitesimal Hopf algebras leads to the computation of the cohomology of ribbon graphs (cf. M. Markl and A.A. Voronov [MV]). In these cases, the homology groups are given by the associated Koszul dual copr- operad. So we get a complete description of them as S-bimodules. In a different License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A KOSZUL DUALITY FOR PROPS 4867 context, S. Merkulov recently gave another proof of the famous formality theorem of M. Kontsevich using the properties of chain complexes of Koszul props in [Me]. In the last section of this paper, we generalize the Poincar´e series of associative algebras and operads to properads. When a properad P is Koszul, we prove a functional equation between the Poincar´eseriesofP and the Poincar´eseriesofits Koszul dual P¡. Conventions Throughout the text, we will use the following conventions. We work over a field k of characteristic 0. 0.1. n-tuples. To simplify the notations, we often write ¯ı an n-tuple (i1, ..., in). The n-tuples considered here are n-tuples of non-negative integers. We denote the sum i1 + ···+ in by |¯ı|. When there is no ambiguity about the number of terms, the n-tuple (1, ..., 1) is denoted by 1.¯ We use the notation ¯ı to represent the “products” of elements indexed by the n-tuple (i1, ..., in). For instance, in the case of k-modules, V¯ı corresponds to ⊗ ···⊗ Vi1 k k Vin . 0.2. Partitions. A partition of an integer n is an ordered increasing sequence (i1, ..., ik) of non-negative integers such that i1 + ···+ ik = n. We denote the set {1, ..., n} by [n]. A partition of the set [n]isaset{I1, ..., Ik} of disjoint subsets of [n] such that I1 ∪···∪Ik =[n]. 0.3. Symmetric group and block permutations. We denote the symmetric group of permutations of [n]bySn . Every permutation σ of Sn is written with S ×···×S S S the n-tuple (σ(1), ..., σ(n)). We denote the subgroup i1 in of |¯ı| by ¯ı. From every permutation τ of Sn and every n-tuple ¯ı =(i1, ..., in), one associates a permutation τ¯ı of S|¯ı|, called a block permutation of type ¯ı, defined by ··· −1 ··· −1 τ¯ı = τi1,..., in := (i1 + + iτ (1)−1 +1, ..., i1 + + iτ (1), ..., i1 + ···+ iτ −1(n)−1 +1, ..., i1 + ···+ iτ −1(n)). 0.4. Gebra. Following the article of J.-P. Serre [S], we call a gebra any algebraic structure like algebras, coalgebras, bialgebras, etc. 1. Composition products In this section we describe an algebraic framework which is used to represent the operations with multiple inputs and multiple outputs acting on algebraic structures. We introduce composition products which correspond to the compositions of such operations. 1.1. S-bimodules. S S P Definition ( -bimodule). An -bimodule is a collection ( (m, n))m, n∈N,ofk- modules P(m, n) endowed with an action of the symmetric group Sm on the left andanactionofthesymmetricgroupSn on the right such that these two actions are compatible. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4868 BRUNO VALLETTE A morphism between two S-bimodules P, Q is a collection of morphisms fm, n : P(m, n) →Q(m, n) which commute with the action of Sm on the left and with the action of Sn on the right.
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